Are there any tricks for strumming during chord transitions?

I am a beginner learning a song with a G-C-D chord sequence on acoustic. I've got the transition down but now I am struggling with the strumming.

The chords switch from 6 strings to 5 strings to 4. When I learned these chords I learned them without muting the strings with my thumb because I just didn't have the dexterity at the time. So I play them by skipping the appropriate strings when needed. But this has gotten difficult when switching between chords and trying to strum at tempo. I keep hitting 6 strings when playing a C and 5 or 6 when playing the D.

So I'm looking for advice. Should I learn to play the chords with muting the extra strings? Or are there any strumming tips/tricks/exercises I could try?

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